Corner and valley conjecture generally refers to hail conjecture. Hail conjecture refers to: a natural number x, if it is odd, multiply it by 3 and add 1, if it is even, it will precipitate an even factor of 2? After many twists and turns, I finally returned to 1.

No matter how great the value in this process is, it falls like a waterfall. Even if other numbers are not, after several transformations, it will inevitably move towards a pure even cycle: 16-8-4-2- 1. According to the research of Japanese and American mathematicians, all natural numbers less than 7 * 10 1 1 conform to this law.

Extended data:

1, the origin of the hail conjecture

All natural numbers are sucked into the black hole (4,2, 1, 4) in a spiral way, and then emitted by the ray (4,2, 1, 4). 1976 One day, Washington Post reported a math news on the front page. This article tells a story: In the mid-1970s, on the campuses of famous American universities, people were frantically playing a math game day and night. The game is simple: write a natural number n at will and transform it according to the following rules:

If it is odd, the next step becomes 3N+ 1. If it is an even number, the next step becomes N/2. Not only students but also teachers, researchers, professors and pedants have joined in.

Why is the charm of this game enduring? Because people find that no matter what number n is, it can't escape and return to the bottom of 1. Accurately speaking, we can't escape the 4-2- 1 cycle that fell to the bottom, and we will never escape such a fate. This is the famous "hail conjecture".

2. Qiang 27

The normalized step number 27 has to undergo many odd-even transformations, and the biggest charm of hail with uneven sawtooth path lies in its unpredictability. John conway, a professor at Cambridge University in England, discovered a natural number of 27. Although 27 is an unremarkable natural number, its ups and downs are extremely severe if the operation is carried out according to the above method:

First, 27 reaches the peak value of 9232 after 77 steps of transformation, and then it reaches the bottom value 1 after 34 steps. The whole transformation process (called "hail process") needs11step, and its peak value is 9232, which is more than 342 times of the original number of 27. Compared with the waterfall-like straight-line falling (2 to the n power), the number n of the same hail process will reach 2 165438. How amazing the contrast is!

However, in the range of 1 to 100, there is no violent fluctuation like that of 27 (except for numbers like 54 which are multiples of 2 of 27).

Baidu encyclopedia-corner valley conjecture